jee-main

Papers (169)
2025
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2024
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2023
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2022
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2021
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2020
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2019
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2018
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
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2016
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2015
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2014
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2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
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2011
jee-main_2011.pdf 13
2010
jee-main_2010.pdf 1
2009
jee-main_2009.pdf 1
2008
jee-main_2008.pdf 1
2007
jee-main_2007.pdf 38
2005
jee-main_2005.pdf 19
2004
jee-main_2004.pdf 11
2003
jee-main_2003.pdf 9
2002
jee-main_2002.pdf 8
2015 04apr

29 maths questions

Q61 Sequences and series, recurrence and convergence Direct term computation from recurrence View
Let $\alpha$ and $\beta$ be the roots of equation $x ^ { 2 } - 6 x - 2 = 0$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n } , \forall n \geq 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } }$ is equal to
(1) - 3
(2) 6
(3) - 6
(4) 3
Q62 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
A complex number $z$ is said to be unimodular if $| z | = 1$. Let $z _ { 1 }$ and $z _ { 2 }$ are complex numbers such that $\frac { z _ { 1 } - 2 z _ { 2 } } { 2 - z _ { 1 } \bar { z } _ { 2 } }$ is unimodular and $z _ { 2 }$ is not unimodular, then the point $z _ { 1 }$ lies on a
(1) circle of radius $\sqrt { 2 }$
(2) straight line parallel to $x$-axis
(3) straight line parallel to $y$-axis
(4) circle of radius 2
Q63 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of integers greater than 6000 that can be formed, using the digits $3,5,6,7$ and 8 , without repetition is
(1) 72
(2) 216
(3) 192
(4) 120
Q64 Geometric Probability View
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $( 0,0 ) , ( 0,41 )$ and $( 41,0 )$ is
(1) 780
(2) 901
(3) 861
(4) 820
Q65 Combinations & Selection Subset Counting with Set-Theoretic Conditions View
Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$, each having at least three elements is
(1) 510
(2) 219
(3) 256
(4) 275
Q66 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
The sum of first 9 terms of the series $\frac { 1 ^ { 3 } } { 1 } + \frac { 1 ^ { 3 } + 2 ^ { 3 } } { 1 + 3 } + \frac { 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } } { 1 + 3 + 5 } + \ldots$ is
(1) 192
(2) 71
(3) 96
(4) 142
Q67 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ are three geometric means between $l$ and $n$, then $G _ { 1 } ^ { 4 } + 2 G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 }$ equals
(1) $4 l ^ { 2 } m ^ { 2 } n ^ { 2 }$
(2) $4 l ^ { 2 } m n$
(3) $4 l m ^ { 2 } n$
(4) $4 l m n ^ { 2 }$
Q68 Generalised Binomial Theorem View
The sum of coefficients of integral powers of $x$ in the binomial expansion of $( 1 - 2 \sqrt { x } ) ^ { 50 }$ is
(1) $\frac { 1 } { 2 } \left( 2 ^ { 50 } + 1 \right)$
(2) $\frac { 1 } { 2 } \left( 3 ^ { 50 } + 1 \right)$
(3) $\frac { 1 } { 2 } \left( 3 ^ { 50 } \right)$
(4) $\frac { 1 } { 2 } \left( 3 ^ { 50 } - 1 \right)$
Q69 Straight Lines & Coordinate Geometry Locus Determination View
Locus of the image of the point $( 2,3 )$ in the line $( 2 x - 3 y + 4 ) + k ( x - 2 y + 3 ) = 0 , k \in \mathbb{R}$, is a
(1) Circle of radius $\sqrt { 3 }$
(2) Straight line parallel to $x$-axis.
(3) Straight line parallel to $y$-axis.
(4) Circle of radius $\sqrt { 2 }$
Q70 Circles Tangent Lines and Tangent Lengths View
The number of common tangents to the circles $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 18 y + 26 = 0$, is
(1) 4
(2) 1
(3) 2
(4) 3
Q71 Circles Circle-Related Locus Problems View
Let $O$ be the vertex and $Q$ be any point on the parabola, $x ^ { 2 } = 8 y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1 : 3$, then the locus of $P$ is
(1) $x ^ { 2 } = 2 y$
(2) $x ^ { 2 } = y$
(3) $y ^ { 2 } = x$
(4) $y ^ { 2 } = 2 x$
Q72 Circles Area and Geometric Measurement Involving Circles View
The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$, is
(1) 27
(2) $\frac { 27 } { 4 }$
(3) 18
(4) $\frac { 27 } { 2 }$
Q73 Sign Change & Interval Methods View
$\lim _ { x \rightarrow 0 } \frac { ( 1 - \cos 2 x ) ( 3 + \cos x ) } { x \tan 4 x } =$
(1) $\frac { 1 } { 2 }$
(2) 4
(3) 3
(4) 2
Q75 Measures of Location and Spread View
The mean of a data set comprising of 16 observations is 16. If one of the observation value 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is
(1) 14.0
(2) 16.8
(3) 16.0
(4) 15.8
Q76 Sine and Cosine Rules Heights and distances / angle of elevation problem View
If the angles of elevation of the top of a tower from three collinear points $A , B$ and $C$ on a line leading to the foot of the tower are $30 ^ { \circ } , 45 ^ { \circ }$ and $60 ^ { \circ }$ respectively, then the ratio $AB : BC$, is
(1) $2 : 3$
(2) $\sqrt { 3 } : 1$
(3) $\sqrt { 3 } : \sqrt { 2 }$
(4) $1 : \sqrt { 3 }$
Q77 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
$A = \left[ \begin{array} { c c c } 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array} \right]$ is a matrix satisfying the equation $A A ^ { T } = 9 I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $( a , b )$ is equal to
(1) $( - 2 , - 1 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( 2,1 )$
Q78 3x3 Matrices Linear System Existence and Uniqueness via Determinant View
The set of all values of $\lambda$ for which the system of linear equations: $2 x _ { 1 } - 2 x _ { 2 } + x _ { 3 } = \lambda x _ { 1 }$ $2 x _ { 1 } - 3 x _ { 2 } + 2 x _ { 3 } = \lambda x _ { 2 }$ $- x _ { 1 } + 2 x _ { 2 } = \lambda x _ { 3 }$ has a non-trivial solution,
(1) Contains more than two elements.
(2) Is an empty set.
(3) Is a singleton.
(4) Contains two elements.
Q79 Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
Let $\tan ^ { - 1 } y = \tan ^ { - 1 } x + \tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right)$, where $| x | < \frac { 1 } { \sqrt { 3 } }$. Then a value of $y$ is
(1) $\frac { 3 x + x ^ { 3 } } { 1 + 3 x ^ { 2 } }$
(2) $\frac { 3 x - x ^ { 3 } } { 1 - 3 x ^ { 2 } }$
(3) $\frac { 3 x + x ^ { 3 } } { 1 - 3 x ^ { 2 } }$
(4) $\frac { 3 x - x ^ { 3 } } { 1 + 3 x ^ { 2 } }$
Q80 Differentiation from First Principles View
If the function $g ( x ) = \left\{ \begin{array} { c c } k \sqrt { x + 1 } , & 0 \leq x \leq 3 \\ m x + 2 , & 3 < x \leq 5 \end{array} \right.$ is differentiable, then the value of $k + m$ is
(1) 4
(2) 2
(3) $\frac { 16 } { 5 }$
(4) $\frac { 10 } { 3 }$
Q81 Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View
The normal to the curve $x ^ { 2 } + 2 x y - 3 y ^ { 2 } = 0$, at $( 1,1 )$
(1) Meets the curve again in the fourth quadrant
(2) Does not meet the curve again
(3) Meets the curve again in the second quadrant
(4) Meets the curve again in the third quadrant
Q82 Stationary points and optimisation Determine parameters from given extremum conditions View
Let $f ( x )$ be a polynomial of degree four and having its extreme values at $x = 1$ and $x = 2$. If $\lim _ { x \rightarrow 0 } \left[ 1 + \frac { f ( x ) } { x ^ { 2 } } \right] = 3$, then $f ( 2 )$ is equal to
(1) 4
(2) - 8
(3) - 4
(4) 0
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The integral $\int \frac { d x } { x ^ { 2 } \left( x ^ { 4 } + 1 \right) ^ { \frac { 3 } { 4 } } }$ equals to
(1) $- \left( \frac { x ^ { 4 } + 1 } { x ^ { 4 } } \right) ^ { \frac { 1 } { 4 } } + c$
(2) $\left( \frac { x ^ { 4 } + 1 } { x ^ { 4 } } \right) ^ { \frac { 1 } { 4 } } + c$
(3) $\left( x ^ { 4 } + 1 \right) ^ { \frac { 1 } { 4 } } + c$
(4) $- \left( x ^ { 4 } + 1 \right) ^ { \frac { 1 } { 4 } } + c$
Q84 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The integral $\int _ { 2 } ^ { 4 } \frac { \log x ^ { 2 } } { \log x ^ { 2 } + \log ( 6 - x ) ^ { 2 } } d x$ is equal to
(1) 6
(2) 2
(3) 4
(4) 1
Q85 Areas by integration View
The area (in sq. units) of the region described by $\left\{ ( x , y ) : y ^ { 2 } \leq 2 x \text{ and } y \geq 4 x - 1 \right\}$ is
(1) $\frac { 9 } { 32 }$ sq. units
(2) $\frac { 7 } { 32 }$ sq. units
(3) $\frac { 5 } { 64 }$ sq. units
(4) $\frac { 15 } { 64 }$ sq. units
Q86 First order differential equations (integrating factor) View
Let $y ( x )$ be the solution of the differential equation $( x \log x ) \frac { d y } { d x } + y = 2 x \log x , ( x \geq 1 )$. Then $y ( e )$ is equal to
(1) $2 e$
(2) $e$
(3) 0
(4) 2
Q87 Vectors Introduction & 2D Angle or Cosine Between Vectors View
Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that no two of them are collinear and $( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }$. If $\theta$ is the angle between vectors $\vec { b }$ and $\vec { c }$, then a value of $\sin \theta$ is
(1) $\frac { - 2 \sqrt { 3 } } { 3 }$
(2) $\frac { 2 \sqrt { 2 } } { 3 }$
(3) $\frac { - \sqrt { 2 } } { 3 }$
(4) $\frac { 2 } { 3 }$
Q88 Vectors: Lines & Planes Find Intersection of a Line and a Plane View
The distance of the point $( 1,0,2 )$ from the point of intersection of the line $\frac { x - 2 } { 3 } = \frac { y + 1 } { 4 } = \frac { z - 2 } { 12 }$ and the plane $x - y + z = 16$, is
(1) 13
(2) $2 \sqrt { 14 }$
(3) 8
(4) $3 \sqrt { 21 }$
Q89 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
The equation of the plane containing the line of intersection of $2 x - 5 y + z = 3 ; x + y + 4 z = 5$, and parallel to the plane, $x + 3 y + 6 z = 1$, is
(1) $2 x + 6 y + 12 z = - 13$
(2) $2 x + 6 y + 12 z = 13$
(3) $x + 3 y + 6 z = - 7$
(4) $x + 3 y + 6 z = 7$
Q90 Discrete Probability Distributions Binomial Distribution Identification and Application View
If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is
(1) $22 \left( \frac { 1 } { 3 } \right) ^ { 11 }$
(2) $\frac { 5 } { 19 }$
(3) $55 \left( \frac { 2 } { 3 } \right) ^ { 10 }$
(4) $220 \left( \frac { 1 } { 3 } \right) ^ { 12 }$